Precalculus Examples

$2 {(8^{x + 1})}^{3} = 64$

Step 1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 \cdot 8^{(x + 1) \cdot 3} = 64$

Step 2

Rewrite $8$ as $2^{3}$ .

$2 \cdot {(2^{3})}^{(x + 1) \cdot 3} = 64$

Step 3

Multiply the exponents in ${(2^{3})}^{(x + 1) \cdot 3}$ .

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Step 3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 \cdot 2^{3 ((x + 1) \cdot 3)} = 64$

Step 3.2

Apply the distributive property.

$2 \cdot 2^{3 (x \cdot 3 + 1 \cdot 3)} = 64$

Step 3.3

Move $3$ to the left of $x$ .

$2 \cdot 2^{3 (3 \cdot x + 1 \cdot 3)} = 64$

Step 3.4

Multiply $3$ by $1$ .

$2 \cdot 2^{3 (3 x + 3)} = 64$

Step 3.5

Apply the distributive property.

$2 \cdot 2^{3 (3 x) + 3 \cdot 3} = 64$

Step 3.6

Multiply $3$ by $3$ .

$2 \cdot 2^{9 x + 3 \cdot 3} = 64$

Step 3.7

Multiply $3$ by $3$ .

$2 \cdot 2^{9 x + 9} = 64$

Step 4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{1 + 9 x + 9} = 64$

Step 5

Add $1$ and $9$ .

$2^{9 x + 10} = 64$

Step 6

Create equivalent expressions in the equation that all have equal bases.

$2^{9 x + 10} = 2^{6}$

Step 7

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$9 x + 10 = 6$

Step 8

Solve for $x$ .

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Step 8.1

Move all terms not containing $x$ to the right side of the equation.

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Step 8.1.1

Subtract $10$ from both sides of the equation.

$9 x = 6 - 10$

Step 8.1.2

Subtract $10$ from $6$ .

$9 x = - 4$

Step 8.2

Divide each term in $9 x = - 4$ by $9$ and simplify.

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Step 8.2.1

Divide each term in $9 x = - 4$ by $9$ .

$\frac{9 x}{9} = \frac{- 4}{9}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $9$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{- 4}{9}$

Step 8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{9}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Move the negative in front of the fraction.

$x = - \frac{4}{9}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{4}{9}$

Decimal Form:

$x = - 0.\overline{4}$